Calculus with Parametric Equations

Slides:

Advertisements

Similar presentations

Unit 6 – Fundamentals of Calculus Section 6

Advertisements

Parametric Equations t x y

Copyright © Cengage Learning. All rights reserved.

Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.

Integral Calculus One Mark Questions. Choose the Correct Answer 1. The value of is (a) (b) (c) 0(d)  2. The value of is (a) (b) 0 (c) (d) 

Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region.

Calculus with Polar Coordinates Ex. Find all points of intersection of r = 1 – 2cos θ and r = 1.

Chapter 8 Plane Curves and Parametric Equations. Copyright © Houghton Mifflin Company. All rights reserved.8 | 2 Definition of a Plane Curve.

10.2 – 10.3 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations.

Polar Coordinates Rectangular (Cartesian) coordinates plot a point by moving left/right and up/down (making a rectangle) Polar coordinates find the.

PARAMETRIC FUNCTIONS Today we will learn about parametric functions in the plane and analyze them using derivatives and integrals.

Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson.

7.4 Length of a Plane Curve y=f(x) is a smooth curve on [a, b] if f ’ is continuous on [a, b].

Applying Calculus Concepts to Parametric Curves 11.2.

CHAPTER Continuity Arc Length Arc Length Formula: If a smooth curve with parametric equations x = f (t), y = g(t), a  t  b, is traversed exactly.

Section 3 Arc Length. Definition Let x = f(t), y = g(t) and that both dx/dt and dy/dt exist on [t 1,t 2 ] The arc length of the curve having the above.

Arc Length and Curvature

How big is my heart??? (Find the area of the enclosed region) WARM UP - Calculator active.

8.1 Arc Length and Surface Area Thurs Feb 4 Do Now Find the volume of the solid created by revolving the region bounded by the x-axis, y-axis, and y =

7.2 Volume: The Disc Method The area under a curve is the summation of an infinite number of rectangles. If we take this rectangle and revolve it about.

10.3 Parametric Arc Length & Area of a Surface of Revolution.

Clicker Question 1 If x = e 2t + 1 and y = 2t 2 + t, then what is y as a function of x ? – A. y = (1/2)(ln 2 (x – 1) + ln(x – 1)) – B. y = ln 2 (x – 1)

Section 9.2: Parametric Equations – Slope, Arc Length, and Surface Area Slope and Tangent Lines: Theorem. 9.4 – If a smooth curve C is given by the equations.

AP Calculus 3.2 Basic Differentiation Rules Objective: Know and apply the basic rules of differentiation Constant Rule Power Rule Sum and Difference Rule.

9.3: Calculus with Parametric Equations When a curve is defined parametrically, it is still necessary to find slopes of tangents, concavity, area, and.

Arc Length & Surfaces of Revolution (7.4)

Calculate the Surface Area of a Cone and Sphere

Parametric equations Parametric equation: x and y expressed in terms of a parameter t, for example, A curve can be described by parametric equations x=x(t),

Lecture 31 – Conic Sections

Implicit Differentiation

Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

Calculus with Parametric Curves

Parametric Equations and Polar Coordinates

Section 15.4 Partial Derivatives

10 Conics, Parametric Equations, and Polar Coordinates

1.) Set up the integral to find the area of the shaded region

Warm-Up! Find the average value of

Arc Length and Curvature

Motion Along a Line: Vectors

By Kevin Dai, Minho Hyun, David Lu

7.2 Volume: The Disc Method The area under a curve

Part (a) Acceleration is the 2nd derivative with respect to time. We already have dx/dt and dy/dt, so we’ll simply take another derivative of each and.

Arc Length and Surface Area

Arc Length and Surfaces of Revolution

Clicker Question 1 If x = e2t + 1 and y = 2t 2 + t , then what is y as a function of x ? A. y = (1/2)(ln2(x – 1) + ln(x – 1)) B. y = ln2(x – 1) + (1/2)ln(x.

Stuff you MUST know Cold for the AP Calculus Exam

Stuff you MUST know Cold for the AP Calculus Exam

Area Between Two Curves

Stuff you MUST know Cold for the AP Calculus Exam

Volumes of Solids of Revolution

Parametric Equations and Calculus

Presentation: Uma Quizizz: Anna 5n1: Emma

Length of Curves and Surface Area

10 Conics, Parametric Equations, and Polar Coordinates

10.2 Parametric Tangents & Areas

7. Section 8.1 Length of a Curve

8.6 – Parametric Equations and Graphs

ENM Mental Calculus Review

Arc Length and Curvature

AP Calculus AB Day 3 Section 7.2 5/7/2019 Perkins.

12.3: Function Analysis of Parametric Equations

Section 7.2 Day 2 Disk Method

10.2 – Calculus with Parametric Curves

12.5: Vector PVA.

Section 6.3 Day 1 Separation of Variable

Chapter 11 Section 11.1 Parametric, Vector, and Polar Functions

Plane Curves and Parametric Equations

Chapter 4 More Derivatives Section 4.1 Chain Rule.

Area and Arc Length in Polar Coordinates

Presentation transcript:

Calculus with Parametric Equations If x = f (t) and y = g(t), then Ex. Find for the curve given by x = sin t and y = cos t, then find the equation of the line tangent to the curve at

Ex. Find all values of t on [0,2π] where the previous curve has a vertical tangent line.

Ex. Determine the slope and concavity of the curve given by at the point (2,3)

Ex. Set up an integral for the area bounded by the curve and the line y = 2.5.

Arc length:

Ex. Find the length of the curve given by x = ln t, y = t + 1 for 1 ≤ t ≤ 6

Surface Area of Revolution: This was x or y depending on axis This will be x(t) or y(t) depending on axis

Ex. Set up an integral for the surface area if the curve given by x = , y = t + 2 for 0 ≤ t ≤ 4 is revolved about the x-axis.

Pract. For Problems 1-2, consider x = t – sin t, y = 1 – cos t 1. Find the equation of the tangent line where t = . 2. Find the length of the curve for 0 ≤ t ≤ 2π. 3. Show that the surface area of a sphere of radius 1 is S = 4π by parametrizing a semicircle and revolving it.